On using estimates of Lipschitz constants in global optimization

Several authors have proposed estimating Lipschitz constants in global optimization by a multiple of the largest slope (in absolute value) between successive evaluation points. A class of univariate functions is exhibited for which the global optimum will be missed when using such a procedure, even if the multiple is arbitrarily large.Research of the first and third authors was supported by AFOSR Grants 0271 and 0066 to Rutgers University. Research of the second author was supported by NSERC Grant GP0036426 and FCAR Grant 90NC0305.This research was done while the first author was Professor and the third author was Graduate Student at RUTCOR, Rutgers University.     

The dual of a finitely generated multiplication module

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in 1986.     

On finitely generated projective modules and multiplication modules

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in January 1987.     

A penalty optimization technique for a class of regulator problems

This paper describes a function space algorithm for the solution of a class of linear-quadratic optimal control problems.The research of the first author was supported by Senate Research Grant No. 8.187.17, University of Ilorin, Ilorin, Kwara State, Nigeria.The authors thank two anonymous referees for their useful and challenging suggestions and comments which improved the quality of this paper. The authors are also indebted to I. Orisamolu for carrying out the computing work.     

Faithful linear representations and residual finiteness of some products of cycles with one relation

Dortmund, Germany. Novosibirsk, USSR. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 204–206, January–February, 1991.     

Characterizing extreme points of polyhedra an extension of a result by Wolfgang Bühler

Summary This paper reconsiders the characterization given by Bühler admitting convex polyhedra of probability distributions on a finite or countable set which are given by systems of linear inequalities more complex than those considered before.

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Zusammenfassung Diese Arbeit charakterisiert die Ecken konvexer Polyeder von Wahrscheinlichkeiten auf einer höchstens abzählbar unendlichen Menge. Diese Polyeder sind durch komplexere Ungleichungssysteme definiert als die von Bühler betrachteten.

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Part of this work was done during the author's stay at the Statistical Laboratory of the University of California, Berkeley and supported by USPHS grant ESO 1299-18.

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W.J. Bühler, is in no way related to the Wolfgang Bühler, Universität Dortmund, whose results are reconsidered here.     

Carleson Measures via BMO

We obtain new characterizations of Carleson measures via uniform boundedness of BMO norms of certain mass functions associated with the given measure in a natural way. This research was performed during M. Stessin’s visit to Korea University. He thanks the Mathematics Department of Korea University and the “Brain Pool” program for their hospitality and support. The first two authors were supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00012).     

Embedding weakly compact sets into Hilbert space

We give an example of a weakly compact set in a Banach space, which does not embed topologically as a weakly compact subset of Hilbert space. We also show that a weakly compact set embeds in a super-reflexive space iff it embeds in Hilbert space. This work was done while the authors were visiting Ohio State University. We wish to thank the Department of mathematics at Ohio State University for their kind hospitality. The work of the first author was partially supported by NSF Grant MPS-74-24249. The work of the second author is a part of his Ph.D. Thesis prepared at the University of California at Berkeley.     

Closed-form control oriented model of highly flexible manipulators

This paper first presents a highly flexible 3D manipulator with links of arbitrary shape, then develops a closed-form dynamic model that best describes it. The model is based on a Newton–Euler formulation and the substructuring method is used to account for large deformations. The formulation of the motion equations starts from a data set which can be either analytically or numerically computed by finite elements(FE) codes. Simulation has been used to validate the model and compare the results with those of two different multibody software and one experimental, which was obtained from the Multi-Elastic-Link Robot Identification Dataset (MERIt), developed by the TU Dortmund. Then, thanks to the approach here adopted, an integral manifold model is derived, suitable for advanced control system design.     

Multimodality analysis of a class of multistage allocation problems

Multiple optimum solutions of a multistage allocation problem, well-known to chemical engineers, are analyzed. The number of local optima becomes greater with a decrease in the initial-condition value of the first stage or with an increase in the total stage number. The fact that this behavior is closely related to the flat portion of the profile of a curvef(x), which determines the objective function, is revealed. A construction method by Aris is used to give an excellent insight into this behavior. Moreover, the construction curves ensure that all stationary points are found. Finally, a theorem to discriminate local optima from stationary points, without evaluating second-order derivatives, is presented.The authors would like to thank Dr. I. Hashimoto and Dr. H. Nishitani for valuable discussions. Computations were carried out with the assistance of Messrs. H. Unno, H. Nakano, Y. Era, and Y. Ueno. The authors are indebted to the computing centers of Osaka University, Kyoto University, and Nagoya University for the use of their facilities.     

Global Optimization Method for Solving Mathematical Programs with Linear Complementarity Constraints

We propose a method for finding a global optimal solution of programs with linear complementarity constraints. This problem arises for instance in bilevel programming. The main idea of the method is to generate a sequence of points either ending at a global optimal solution within a finite number of iterations or converging to a global optimal solution. The construction of such sequence is based on branch-and-bound techniques, which have been used successfully in global optimization. Results on a numerical test of the algorithm are reported.The main part of this article was written during the first authors stay as Visiting Professor at the Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Japan. The second and the third authors were supported by Grant-in-Aid for Scientific Research C(2) 13650061 of the Ministry of Education, Culture, Sports, Science, and\break Technology of Japan.The authors thank P. B. Hermanns, Department of Mathematics, University of Trier, for carrying out the numerical test reported in Section 5. The authors also thank the referees and the Associate Editor for comments and suggestions which helped improving the first version of this article.     

Two topological properties of topological linear spaces

A topological and a geometrical-topological property, previously known only for normed linear spaces, are established here for much more general classes of topological linear spaces. This research was conducted at the University of Washington in 1963 when the first author was visiting there. The work of both authors was supported in part by the National Science Foundation, U. S. A. (NSF-GP-378).     

Products of conjugacy classes in Chevalley groups II. Covering and generation

In this paper we establish a relationship between generating numbers and covering numbers of conjugacy classes in Chevalley groups over algebraically closed fields. The authors gratefully acknowledge EPSRC grant GR58542. The first author acknowledges further a grant SFB 343 “Diskrete Strukturen in der Mathematik”. The second author thanks the Institute for Advanced Studies at The Hebrew University for its hospitality.     

Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. The authors wish to express their gratitude to B. A. Davey and T. Katriňák for their helpful remarks and to J. G. Pitkethly for her assistance with the pictures. A support by Slovak grants VEGA 1/4057/97, 1/3026/06 and APVV-51-009605 is acknowledged by the first author who also wishes to thank the Mathematical Institute of the University of Oxford and the School of Mathematical and Statistical Sciences of La Trobe University for their hospitality.     

Analysis of heuristics for the design of tree networks

We analyze the tour partitioning heuristics for the Capacitated Minimum Spanning Tree problem. Lower bounds for the worst-case performance ratios of these heuristics are obtained by using worst-case examples. We also generalize the heuristics to the multi-center case with the same worst-case bounds.The work of the first author was supported by a Dean Summer Research Grant from Owen Graduate School of Management, Vanderbilt University.Work done in part in the Department of Industrial Engineering and Operations Research at Columbia University.The work of the last two authors was supported in part by ONR contract N00014-90-J-1649, NSF contract DDM-8922712 and the Center for Telecommunications Research under NSF contract CDR 84-21402.     

A computational study of methods for solving polynomial geometric programs

A computational comparison of several methods for dealing with polynomial geometric programs is presented. Specifically, we compare the complementary programs of Avriel and Williams (Ref. 1) with the reversed programs and the harmonic programs of Duffin and Peterson (Refs. 2, 3). These methods are used to generate a sequence of posynomial geometric programs which are solved using a dual algorithm.The authors would like to acknowledge the helpful comments of the referees. Also, they would like to acknowledge the programming assistance of Mr. S. N. Wong of The Pennsylvania State University. The first author's research was supported in part by a Research Initiation Grant awarded through The Pennsylvania State University.     

Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. This work was initialized during the stay of the first and second named authors at the Jagiellonian University, Kraków in October, 2006, supported by the EGIDE program. They wish to thank both institutions. The third author was supported by the KBN research grant No. 1 PO3A 005 28.     

There exists a maximal 3-C.E. enumeration degree

We construct an incomplete 3-c.e. enumeration degree which is maximal among then-c.e. enumeration degrees for everyn with 3≤n≤ω. Consequently then-c.e. enumeration degrees are not dense for any suchn. We show also that no lown-c.e. e-degree can be maximal among then-c.e. e-degrees, for 2≤n≤ω. The first two authors were partially supported by EPSRC Research Grant “Turing Definability” No. GR/M 91419 (UK), and the second author by NSF grant No. 69973048 and by NSF major grant No. 19931020 (P. R. China), and by an INDAM visiting professorship at the University of Siena. The fourth author was partially supported as a visiting scholar by the University of Siena. The first three authors were funded by the INTAS-RFBR joint projectComputability and Models, no. 972-139. The fourth authors would like to thank Marat Arslanov for useful discussions.     

Dynamic common property resources and environmental problems

Problems of interacting common-property resources are set up as stochastic differential games. A class of models is solved where equilibrium closed-loop strategies keep harvest rates proportional to stocks. Corrective taxes, etc., are considered.Dedicated to G. LeitmannEarlier versions of this paper (Ref. 1) were presented at the International Conference on Economics and Control, Cambridge University, England, June 18–21, 1979, and at the Winter Econometric Society Meetings, Atlanta, Georgia, December 1978. Space limitations shortened this paper; a complete working paper is available from the authors. The authors would like to extend their appreciation to George Leitmann, who first introduced them to differential games.     

Primal-Dual Simplex Method for Multiobjective Linear Programming

We develop a primal-dual simplex algorithm for multicriteria linear programming. It is based on the scalarization theorem of Pareto optimal solutions of multicriteria linear programs and the single objective primal-dual simplex algorithm. We illustrate the algorithm by an example, present some numerical results, give some further details on special cases and point out future research. The paper was written during a visit of the first author to the University of Sevilla financed by a grant of the Andalusian Consejería de Educación. The research of the first author was partially supported by University of Auckland Grant 3602178/9275. The research of the second and third authors was partially financed by Spanish Grants BFM2001-2378, BFM2001-4028, MTM2004-0909 and HA2003-0121. We thank Anthony Przybylski for the implementation and making his results available. We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper.